C₄×C₂₉×#124 #16816017

order	3712
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₄×C₂₉×#124

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂³
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂³	C₂⁴×C₂₉
socle	C₂³×C₂₉	C₂⁴
prime radical	C₂²×D₄×C₂₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ G

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(6,7)(11,15)(13,16);(5,6,8,7)(9,10,12,14)(11,13,15,16);(5,7,8,6)(9,11)(10,13)(12,15)(14,16)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7567, C₂₉×#7584, C₂₉×#7601, …
maximal normal subgroup of	C₂₉×#7567, C₂₉×#7584, C₂₉×#7601, …
maximal quotient of	C₄×C₂₉×#388, C₄×C₂₉×#390, C₄×C₂₉×#391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1035);

C4×C29×#124 #16816017